Diploma Thesis - uni-bielefeld.dekrssak/pdf/Krssak... · 2013-03-19 · work on this thesis, mainly my parents and Olga. Declaration. I declare that I have written my diploma thesis

Charles University in Prague

Faculty of Mathematics and Physics

Diploma Thesis

Martin Krssak

On higher dimensional Kerr-Schild spacetimes

Institute of Theoretical Physics

Supervisor: Dr. Marcello Ortaggio

Branch of Study: Theoretical Physics

Acknowledgment.

I would like to thank to Dr. Marcello Ortaggio for highly valuable help with prepa-

ration of manuscript of this thesis. Beside him I would like to thank to M. Imran

from University of Durham for writing this LATEX template and making it publicly

available.

Also i would like to express my gratitude to people who supported me during

work on this thesis, mainly my parents and Olga.

Declaration.

I declare that I have written my diploma thesis on my own with a help of liter-

ature listed in Bibliography. I agree with lending of my work.

In Prague, 5th of August 2009 Bc. Martin Krssak

Abstrakt.

Nazev prace: O vıcerozmernych Kerr-Schildovych caso-prostorech

Autor: Martin Krssak

Ustav: Ustav teoreticke fyziky

Vedoucı diplomove prace: Dr. Marcello Ortaggio, Matematicky Ustav, AV CR

e-mail vedoucıho: [email protected]

Abstrakt: V tejto diplomovej praci sa venujeme Kerr-Schildovym (KS) casopriesto-

rom vo vyssıch (n > 4) dimenziach s geodetickym nulovym vektorom `. S pouzitım

zovseobecnenia Newman-Penrosoveho formalizmu do vyssıch dimenzii, najdeme, pre

metriku v tvare KS ansatzu, zodpovedajuce Einsteinove rovnice a zameriame sa na

vakuove riesenia.

Spomenieme najnovsie vysledky v prıpade neexpandujucich riesenı a my sa za-

meriame na expandujuce. Zistıme, ze jedna z Einsteinovych rovnıc, ktoru nazyvame

opticka vazba, plne urcuje `. Zvysok Einsteinovych rovnıc urcuje KS funkciu H.

Najdene rovnice analyzujeme a zistıme, ze sa jedna o system nelinearnych PDR

s nelinearnymi vazbami, co nam zabranuje najst analyticke riesenie vo vseobecnom

prıpade. Preto sa musıme uspokojit s ciastocnymi vysledkami.

Zameriame sa na riesenia bez ”twistu”. V ramci tejto triedy riesenı najdeme

vhodny ansatz splnujuci opticku vazbu s pomocou ktoreho sme schopnı najst expli-

citne riesenie vsetkych Einsteinovych rovnıc. Ukazeme, ze taketo riesenie zodpoveda

ciernej strune a ukazeme ako sa da toto riesenie zovseobecnit, tak aby odpovedalo

ciernej ”p-brane”.

Druhym prıstupom je skumanie r-zavislosti KS caso-priestorov. Za r si zvolıme

afinny parameter pozdlz vektora `. Najdeme r-zavislost vsetkych Ricciho koefi-

cientov, KS funkcie H a operatorov smerovych derivacii do lubovolneho radu v r.

S tymito vedomostami sme schopnı najst r-zavislost vsetkych komponent Rieman-

novho tenzora a analyzujeme jeho asymptoticke vlastnosti.

Klıcova slova: Kerr-Schildove caso-prostory, vyssı dimenze, cerne dıry, Newman-

Penrose formalizmus.

Abstract.

Title: On higher dimensional Kerr-Schild spacetimes

Author: Martin Krssak

Department: Institute of Theoretical Physics

Supervisor: Dr. Marcello Ortaggio, Mathematical Institute, AV CR

Supervisor’s e-mail address: [email protected]

Abstract: In this thesis we investigate Kerr-Schild (KS) metrics in higher (n > 4)

dimensions with geodetic null-congruence `, using a generalization of Newman-

Penrose formalism to higher dimensions. We use KS ansatz for metric and find

corresponding Einstein’s equations. We focus on vacuum solutions of Einstein’s

equations.

We give some remarks on recent results about non-expanding solution and we

focus on expanding solutions. One of Einstein’s equation, to which we refer as

the optical constraint, determines completely the null-congruence. The remaining

determine KS function H.

We analyze these equations and find that they are a systems of nonlinear PDEs

with non-linear couplings, what prohibit us from finding analytical solution in gen-

eral case. Therefore, we have to satisfy ourselves with partial results.

We thus focus on non-twisting solutions. Within this class, we find a suitable

ansatz which satisfies the optical constraint and we solve all the corresponding Ein-

stein equations explicitly. We further demonstrate that this solution corresponds

to (static) black strings and we give instruction how to generalize it to the black

p-branes.

Second approach is that we fix r-depependence choosing affine parameter along

null vector `. We derive r-dependance of all Ricci coefficients, KS function H and

operators of directional derivative to arbitrary term in r. Using this knowledge we

find r-dependance of all components of Riemann tensor and analyze their asymptotic

properties.

Keywords: Kerr-Schild solutions, higher dimensions, black holes, Newman-Penrose

formalism.

Contents

Acknowledgement ii

Declaration ii

Abstrakt iii

Abstract iv

1 Introduction 2

1.1 Basic black hole solutions . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Original Kerr-Schild solutions . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Myers-Perry solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Newman-Penrose formalism in higher dimensions 10

2.1 Frame formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Newman-Penrose formalism . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Null frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Ricci rotation coefficients . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Curvature tensor and Ricci tensor . . . . . . . . . . . . . . . . 16

2.2.4 Other issues in NP formalism . . . . . . . . . . . . . . . . . . 17

3 Kerr-Schild spacetimes 18

3.1 Kerr-Schild ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Rotation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Antisymmetric part of rotation coefficients . . . . . . . . . . . 21

v

Contents On higher dimensional Kerr-Schild spacetimes vi

3.2.2 Lab coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.3 Nab coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.4i

M ab coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.5 Condition of geodecity . . . . . . . . . . . . . . . . . . . . . . 23

3.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Connection and curvature forms . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Einstein equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Vacuum Kerr-Schild solutions 31

4.1 Vacuum Einstein’s equations . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Non-expanding solutions . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3 Expanding solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Comments on solving Einstein’s equations . . . . . . . . . . . 35

4.4 Non-twisting solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Some explicit solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.6 Fixing r-dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Conclusion 49

Bibliography 52

Appendix 56

A Notation 56

List of Tables

3.1 Rotation coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Components of Riemann tensor . . . . . . . . . . . . . . . . . . . . . 28

3.3 Ricci tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1

Chapter 1

Introduction

One of the main tasks of theoretical physics during the last 5-6 decades was, and still

remains, to find a theory which would unify all four fundamental forces of Nature.

During 70s, the quantum description of three of these forces was successfully unified

to one theory, presently known as the Standard Model . However, despite of a huge

effort of physicists world-wide it was not possible to incorporate gravitation into the

Standard Model or to find a consistent quantum theory of gravitation. It turned

out that gravitation at sub-atomic scales is very different from all other interactions

and that there is a need for a new theory.

Nowadays, we have basically three main theories which are aspiring to be cor-

rect quantum theory of gravitation. The main one, which dominates today in the-

oretical physics is string theory, which suggests that problems of quantization of

gravitation can be resolved by considering fundamental particles as one-dimensional

objects, i.e. we can think of them as a tiny strings [1]. Second approach, a wide

family of approaches called canonical quantum gravitation, is trying to develop non-

perturbative methods to quantize gravitation [2]. The most famous of these theories

is the loop quantum gravity in which space is represented by quantized loops called

spin networks [3]. Other approach, gaining popularity recently, is non-commutative

geometry [4].

Our work presented in this thesis is directly in neither of these theories, but in a

field of classical gravitation in spacetimes with more than 4 dimensions. In order to

understand motivation for such work, we will mention main features of string theory

2

Chapter 1. Introduction On higher dimensional Kerr-Schild spacetimes 3

and how it is connected with matter of our research.

One of the main flaws of Standard Model, beside its lack of description of grav-

itation, is that it has 19 free parameters (masses of 3 leptons, 6 quarks, coupling

constants,...) which are needed to fully describe dynamics of the theory [5]. These

parameters in Standard model can take arbitrary values and have to be find ex-

perimentally. However, this is a very inelegant feature of Standard Model, because

it seems that if these free-parameters would have just slightly different values, our

Universe would be very different from what we are presently experiencing and would

not allow creation of galaxies, star systems, stable atoms and support creation of

life.

String theory solves this problem elegantly, because the only free parameter of

this theory is string tension and all particles are considered to be just different

vibrational modes of fundamental string. Beside claiming of explanation of masses

of fundamental particles, there was found spin-two field representing gravitation.

Thus string theory is claiming to be a consistent quantum theory of gravitation.

However, to make this theory consistent we need to introduce supersymmetry

and 6 extra dimensions, i.e. spacetime is required to be 10 dimensional. The two

most popular ways how to deal with extra-dimensions are compactification of extra-

dimensions and brane models. The last one suggests that our universe lives on 4

dimensional brane floating in 10 dimensional space, where 3 fundamental interac-

tions of Standard Model are confined to this brane, while gravitation acts in full

10 dimensional space. This solves so-called hierarchy problem, i.e. explains relative

weakness of gravitation against other interactions [6].

Idea of compactification of extra-dimensions suggests that the reason why we

cannot see these extra dimensions is that they are such small that they are unde-

tectable by ordinary experiments. Scale of these extra-dimensions is not determined

by string theory and theoretically can be as small as the Planck length, what would

make any laboratory test of string theory unrealistic in any foreseeable future, or,

as it suggests so-called large dimensions scenarios, can be sufficiently large to make

string theory testable at LHC which should start to produce the first measurements

soon.

Chapter 1. Introduction On higher dimensional Kerr-Schild spacetimes 4

If the scale of extra-dimensions is large enough we could be able to observe

microscopic black-holes at the LHC, see [7], [8], [9], [10] or recent review [11]. This

would not allow us to test just string theory, but also to make laboratory observation

of black holes for first time1.

Beside important role in possible experiments, higher dimensional black holes

(more specific their generalizations black p-branes) are important in investigation of

non-perturbative effect of string theory [1], [13]. Also in gauge/string duality known

as AdS/CFT [14], [15], solutions of higher dimensional gravitation can be useful.

But, first we need to better understand classical black hole solutions in higher

dimensions , in order to understand which effects are specific to string theory and

which are purely classical gravitational effects. Thus, string theory is an impor-

tant source of motivation to understand solutions of classical gravitation in higher

dimensions.

Beside, interesting results for string theory or possible experiments, classical

gravitation in higher dimensions is interesting on its own. For example, famous

topology theorem by Hawking [16], [17] that a stationary black hole horizon has

always the spherical topology does not hold in higher dimensions. Solutions as

black strings, or recently found black rings [18] are well-known counter-examples.

Thus, question of uniqueness of black holes in higher dimensions is much more

complicated, see [21] and [20] for recent progress.

Thus, research in higher dimensional gravitation is interesting from many per-

spectives. In this thesis we will focus on class of solutions called Kerr-Schild solu-

tions, which played important role in four dimensional gravitation and we will try

to analyze some of their properties in higher dimensions.

Let us first give very brief outlook of the most important results about black

holes in higher dimensions before we start to analyze Kerr-Schild spacetimes.

1Note, that we would not observe directly black holes, but analogue of Hawking radiation from

these black holes. Hawking radiation was already experimentally observed in solid states [12], so

we may expect to occur in a gravitational case and thus saving us from growing black hole to

dangerous size [11].

1.1. Basic black hole solutions On higher dimensional Kerr-Schild spacetimes 5

1.1 Basic black hole solutions

General relativity describes dynamics of space-time. Fundamental equations of this

theory are Einstein’s equations, which in a natural units ~ = G = c = 1 and without

cosmological constant have a form

Rab − 1

2gabR = 8πTab, (1.1)

i.e. they say how geometry of spacetime (left-hand side of these equations) is coupled

to matter (described by right-hand side of these equations). Dynamical variable of

Einstein’s equations is metric tensor gab, which is a symmetric rank-two tensor, i.e.

it has generally n(n+1)2

independent components. Thus, to solve Einstein’s equations

(1.1) means to find all independent components of the metric tensor. Our work in

whole thesis will be restricted to the case of vacuum space-times2, where geometry

of space-time is given by vacuum Einstein’s equations

Rab = 0. (1.2)

Even vacuum Einstein’s equations are a system of non-linear partial differential

equations and there exists no general analytical solution. However, it is possible

to find some solutions if one assumes that metric tensor possess symmetry. In the

case of spherical symmetry we are able to find that the most general spherically

symmetric metrical tensor should have a following form [17]

ds2 = −f(t, r)dt2 + g(t, r)dr2 + r2dΩ2, (1.3)

where

dΩ2 = dθ2 + sin2 θdφ2. (1.4)

If we insert this metric into the vacuum Einstein’s equations we find well-known

Schwarzschild solution

ds2 = −(

1− 2M

r

)dt2 +

dr2

(1− 2M

r

) + r2dΩ2. (1.5)

2However, we will occasionally mention generalizations containing simple form of energy-

momentum, like electromagnetic field, or cosmological constant, and we will give corresponding

references.

1.2. Original Kerr-Schild solutions On higher dimensional Kerr-Schild spacetimes 6

Generalization of this solution to arbitrary dimension is quite straightforward and

was found by Tangherlini in 1963 [22]

ds2 = −(

1− 2M

rn−3

)dt2 +

dr2

(1− 2M

rn−3

) + r2dΩ2n−2, (1.6)

this solution is usually referred as a Scharzschild-Tangherlini solution.

If we impose less-restrictive symmetry we are able to derive solution representing

rotating black hole. In four dimensions black hole can rotate just in a single plane,

however in higher dimensions situation is different and we can have bn−12c planes of

rotation [23], where b· · · c represents integer part of a given number.

Solution with a single plane of rotation is given by [23], [24]

ds2 = −dt2 +M

rn−5Σ(dt− a sin2 θdφ)2 +

Σ

∆dr2 + Σdθ2

+ (r2 + a2) sin2 θdφ2 + r2 cos2 θdΩ2n−4, (1.7)

where

Σ = r2 + a2 cos2 θ, ∆ = r2 + a2 − M

rn−5. (1.8)

In the case n = 4 this is well-known Kerr solution [25], which is one of most

important known solution in general relativity. Generalization of this solution with

more spins are Myers-Perry solutions [23], which will be briefly described in section

1.3. Other black hole solution in higher dimensional gravitation are black strings

and generally black p-branes [13], black rings [18] and many other solutions, for

example solutions with supersymmetry, or cosmological constant. We refer reader

to reviews [24], [19] for more details.

1.2 Original Kerr-Schild solutions

In 1960s Kerr, et. al. [26], [27], [28], [29] found, in four dimensions, general class

of solutions which contains Kerr solutions as special case. This class of solutions

is presently known as Kerr-Schild spacetimes. In this approach we do not demand

spherical symmetry or axisymmetry of metric tensor, but we use following ansatz

for a metric

gab = ηab + 2Hkakb, (1.9)

1.2. Original Kerr-Schild solutions On higher dimensional Kerr-Schild spacetimes 7

where function H is some analytic function and ηab is metric of a four-dimensional

Minkowski space-time and ka is a congruence of null vectors with respect to both

metrics gab and ηab

gabkakb = ηabk

akb = 0. (1.10)

It is usefull to use null coordinates

u = 2−12 (z + t), ζ = 2−

12 (x+ iy),

v = 2−12 (z − t), ζ = 2−

12 (x− iy).

Most general null direction in such coordinates is given by

k = du+ Y dζ + Y dζ − Y Y dv, (1.11)

where Y is some analytic function of coordinates and Y its complex conjugate.

Advantage of using complex function and coordinates is that then we have just half

of necessary equation, because other half can be obtained by complex conjugation.

General solution of Einstein’s equations in such case exists and is given by

ds2 = 2dζdζ + 2dudv

+ P−3[m(Z + Z)− ψψZZ

] (du+ Y dζ + Y dζ − Y Y dv

)2, (1.12)

where

P = pY Y + qY + qY + c, (1.13)

Z = −PF−1Y , (1.14)

and Y is given implicitly by

F = 0, (1.15)

and F is defined as

F = φ+ (qY + c)(ζ − Y v)− (pY + q)(u+ Y ζ), (1.16)

where φ and ψ are arbitrary analytic functions of the complex variable Y and m, p,

c are real constants and q is complex constant.

Let us note, that equations (1.15) and (1.16) fully determine Y and consequently

fully determine the null-congruence ka. This is known as the Kerr theorem and in

1.3. Myers-Perry solutions On higher dimensional Kerr-Schild spacetimes 8

four dimensions its solution is the most general shear-free, geodetic congruence in

flat space [27], [30]. This is an important result not only in investigations of Kerr-

Schild spacetimes, but have applications also in other field of physics, for example

twistor theory [30].

1.3 Myers-Perry solutions

In 1986 Myers and Perry [23] used generalization of the Kerr metric [25] and thus

obtained one of very few known Kerr-Schild solutions in higher dimensions repre-

senting black hole rotating in many planes of rotation.

As it was mentioned earlier, black holes in higher dimensions can rotate in bn−12c

plane of rotation and thus it is convenient to pair the coordinates as xa = xα, yα,where α = 1, · · · bn−1

2c. Consequence of this is that Myers-Perry black holes have

separate solutions for odd and even dimensional spacetimes. Let us start with odd

dimensions. In this case Kerr-Schild metric (1.9) is fully determined by

k = kadxa = dt+

bn−22c∑

α=1

r(xαdxα + yαdyα) + aα(xαdyα − yαdxα)

r2 + aα2

, (1.17)

and

H =µr2

ΠF, (1.18)

where aα are some constants and r is defined implicitly by

bn−22c∑

α=1

xα2 + yα2

r2 + aα2

= 1, (1.19)

and

F = 1−bn−2

2c∑

α=1

aα2(xα2 + yα2)

(r2 + aα2)2

, (1.20)

Π =

bn−22c∏

α=1

(r2 + aα2). (1.21)

In even dimensions the solution is given by

k = kadxa = dt+

bn−22c∑

α=1

r(xαdxα + yαdyα) + aα(xαdyα − yαdxα)

r2 + aα2

+zdz

r, (1.22)

1.4. Outline On higher dimensional Kerr-Schild spacetimes 9

and r is defined implicitly by

bn−22c∑

α=1

xα2 + yα2

r2 + aα2

+z2

r2= 1, (1.23)

with F and Π as in previous case. We can easily check that in 4 dimensions this is

equivalent to the Kerr metric in coordinate basis [25]. There exist generalizations

of Myers-Perry solution to the case of dS and AdS spacetimes [31].

1.4 Outline

Generalization of these results to the higher dimensions will be our main interest

in this thesis. We will see that this generalization is not straightforward at all and

that we will encounter many problems. Situation indeed is very different from four

dimensional case because we have many degrees of freedom.

As a illustration take the Kerr theorem, mentioned at the end of section 1.2. In

four dimensions introducing complex function Y allowed us to specify null congru-

ence k uniquely by this single function. Consequence of this was that null congruence

k was fully determined by simple partial differential equation of first order, which

we know how to solve analytically.

As we will see in following chapters, in higher dimension our null congruence

will be specified by n − 2 functions and resulting equations will be system of non-

linearly coupled partial differential equations. However, we will try to analyze these

equations and try to find their solutions at least in some special cases.

We will start in chapter 2 with introduction of Newman-Penrose formalism in

higher dimensions and we will summarize most important results which we will need

in following calculations.

We will continue in chapter 3 with adopting Kerr-Schild ansatz metric (1.9) in

Newman-Penrose formalism and we will find Einstein’s equations for it.

In chapter 4 we will analyze Einstein’s equations, we will give reasons why we

have to give up hope to solve them analytically in general case and we solve them

at least in some special cases.

Chapter 2

Newman-Penrose formalism in

higher dimensions

In investigation of Kerr-Schild spacetimes we will use generalization of Newman-

Penrose (NP) formalism [32] in higher dimensions. In this chapter we will give

introduction to tetrad formalism and to Newman-Penrose formalism.

2.1 Frame formalism

At each point of space-time it is possible to introduce n-tuple of n independent

vectors m ba , where normal letters are tensor indices and indices with hat are frame

indices and label the different vectors of the n-tuple. Both kinds of indices run from

0 to n− 1, but we always have to keep in mind that they are two different kinds of

indices. Einstein summation convention holds for both kinds of indices.

This n-tuple of vectors in four dimensions is usually called tetrad or, what is

more appropriate in higher dimensions, frame, from where we have name for the

whole formalism- frame formalism.

Dual frame is defined by either of the equivalent relations

m ca m

bc = δb

a, m ba m

ac = δb

c. (2.1)

Using this we can find the frame components of any tensor T b···a··· from relation

T b···a··· = m c

a mbd · · ·T d···

c··· , (2.2)

10

2.1. Frame formalism On higher dimensional Kerr-Schild spacetimes 11

and inverse relation is given by

T b···a··· = mc

amb

d· · ·T d···

c··· . (2.3)

The frame vectors determine differential forms

ma = mabdx

b, (2.4)

and their duals

ma = m ba ∂b. (2.5)

So, the metric form will be given by

ds2 = mama = gabdx

adxb. (2.6)

The directional derivative of function along a frame vector will be denoted by |a or

by δa

f|a = δaf = m ba

∂f

∂xb. (2.7)

The frame components of a covariant derivative are projections of coordinate co-

variant derivatives

T b···a··· ;c = T e···

d··· ;gmg

c md

a mbe · · · , (2.8)

and they are given by

T b···a··· ;c ≡ T b···

a··· |c − ΓdacT

b···d··· − · · ·+ Γb

dcT d···

a··· + · · · , (2.9)

where Γabc

are Ricci rotation coefficients

Γabc

= −mad;em

dbm e

c , (2.10)

we can lower first index to obtain

Γabc = gadΓdbc. (2.11)

We can define new 1-forms called connection forms, which are projections of Ricci

rotation coefficients on frame vectors

Γab= Γa

bcmc , (2.12)

2.2. Newman-Penrose formalism On higher dimensional Kerr-Schild spacetimes 12

and curvature form, which are projections of frame components of curvature tensor

on frame vectors

Rab = Rabcdmc ∧md . (2.13)

Both of them can be obtained from frame forms using Cartan equations

dma = mb ∧ Γab= Γa

bcmb ∧mc , (2.14)

1

2Ra

b= dΓa

b+ Γa

c ∧ Γcb. (2.15)

Equation (2.14) determines just skew-symmetric part of the Ricci coefficients Γa[bc].

The symmetric part Γ(ab)c of the Ricci coefficients is given by

2Γ(ab)c = gab|c, (2.16)

since

0 = gab;c = gab|c − 2Γ(ab)c. (2.17)

In all calculations in this thesis we will restrict just to the case of rigid frames, where

gab are constants and thus from (2.16) we can find that Ricci rotation coefficients are

skew-symmetric in the first two indices. Thus they can be determined from (2.14)

by

Γabc = −Γbac = Γa[bc] + Γb[ca] − Γc[ab]. (2.18)

For more details and derivations see [27], [29] or [33].

2.2 Newman-Penrose formalism

Special form of tetrad formalism was developed in 1960s by E. Newman and R.

Penrose [32], who used a special form of the frame metric gab. While in tetrad

formalism Minkowski metric, with orthonormal frame vectors, was used, the idea of

Newman and Penrose was to use null metric, which turned out to be very useful,

mainly in the investigation of algebraically special spacetimes (according to Petrov

classification) [17], [29]. One feature of this approach is that we obtain large amount

of equations, from which many are redundant, however advantage of this approach

is that all obtained differential equations are of first order.


2.2.1 Null frames

NP formalism which we will use in the analyzes of Kerr-Schild spacetimes has to be

generalized to higher dimensions. This was done in papers [34], [35] and [36].

We choose our frame metric to be

gab =

0 1 . · · · .

1 0 . · · · .

. . 1 · · · .

......

.... . .

...

. . . · · · 1

, (2.19)

where · · · always represents zero, except diagonal where it represents 1. So, this

means that we are choosing null frame with two null vectors m0 = n and m1 = `

and n− 2 orthonormal space-like vectors mi for which

lala = nana = namia = lami

a = 0,

lana = 1, (2.20)

miamja = δij.

From now on, indices taken from start of alphabet runs a, b, · · · = 0, . . . , n − 1 and

indices from middle of alphabet i, j, · · · = 2, . . . , n− 1.

Lorentz transformations of frame vectors ma can be described by null rotations

(where ` or n are fixed), spatial rotations of the vectors mi (spins) and boosts.

Null rotations:

˜ = `, (2.21)

n = n+ zimi − 1

2z2`, (2.22)

mi = mi − zi`, (2.23)

where zi are some real functions and z2 = z izi.

Spins:

˜ = `, (2.24)

n = n, (2.25)

mi = X ijmj , (2.26)


where X ∈ SO(n− 2).

Boosts:

˜ = λ`, (2.27)

n = λ−1n, (2.28)

mi = mi , (2.29)

where λ is some function λ ∈ R.

2.2.2 Ricci rotation coefficients

We can find Ricci rotation coefficients (2.10) in this null frame. They will give us

information about covariant derivatives of frame vectors. We will define them to be

la;b = Lcdmcam

db ,

na;b = Ncdmcam

db , (2.30)

mia;b =

i

M cdmcam

db .

Note, that in notation used in (2.10) these coefficients are

Nbc = −Γ0bc

= −Γ1bc, (2.31)

Lbc = −Γ1bc

= −Γ0bc, (2.32)

i

M bc = −Γibc

= −Γibc. (2.33)

If we take first derivatives of scalar products of basis vectors from previous section

(2.20) we find

L0a = N1a =i

M ia = 0, (2.34)

N0a + L1a =i

M 0a + Lia =i

M 1a +Nia =i

M ja +j

M ia = 0, (2.35)

which reduces a number of independent rotation coefficients to n2(n− 1)/2.

We define covariant derivatives along the frame vectors in analogy with 4 dimensional

NP formalism, which acts on scalar as

Df = f|0 ≡ la∇af, ∆f = f|1 ≡ na∇af, δif = f|i ≡ mia∇af. (2.36)


Expansion, shear and twist

In four dimensions for complex null frame we can define well-known optical scalars,

namely expansion, shear and twist. In higher dimensions the situation is more

complicated. We will be able to obtain similar quantities as in 4 dimensions, but

with difference that they will be not scalars anymore, they have to be generalized

to vectors or matrices.

Physically interesting case is when the vector ` is geodetic. From (2.30) we can see

that

la;blb = L10la + Li0m

(i)a , (2.37)

so vector ` is geodetic iff Li0 = 0. In a such case, according to transformation

properties of rotation coefficients given in [36], we can find that Lij is invariant

under null rotations and transforms under boosts simply as

Lij = λLij, (2.38)

and thus it has special geometric meaning, because it characterizes null congruence

in an invariant way. The matrix Lij can be decomposed into

Lij = Sij + Aij, (2.39)

where

Sij ≡ L(ij) = σij + θδij, (2.40)

Aij ≡ L[ij], (2.41)

where σij and Aij are shear and twist matrices, respectively. From these we can

define three scalars under space and null rotations θ, σ, ω, which we call expansion,

shear and twist scalars, respectively.

They are given by

θ ≡ 1

n− 2Sii, σ2 ≡ σijσij, ω2 ≡ AijAij, (2.42)

where Einstein summation convention applies.


2.2.3 Curvature tensor and Ricci tensor

In order to solve Einstein’s equations we have to find components of Ricci tensor in

Newman-Penrose formalism. Second Cartan equation (2.15) will be used

dΓab + Γac ∧ Γdbgcd =

1

2Rab. (2.43)

Ricci rotation coefficients are determined from connection forms which are skew-

symmetric, so we have just these independent forms Γ01,Γ0i,Γ1i,Γij:

dΓ01 + Γ0i ∧ Γi1 =1

2R01 =

1

2R01abm

a ∧mb , (2.44)

dΓ0i + Γ01 ∧ Γ0i + Γ0j ∧ Γji =1

2R0i =

1

2R0iabm

a ∧mb , (2.45)

dΓ1i − Γ01 ∧ Γ1i + Γ1j ∧ Γji =1

2R1i =

1

2R1iabm

a ∧mb , (2.46)

dΓij + Γi0 ∧ Γ1j − Γj0 ∧ Γ1i + Γik ∧ Γkj =1

2Rij =

1

2Rijabm

a ∧mb . (2.47)

The Ricci tensor is defined to be

Rab = Rba = Rcacb. (2.48)

It has the following independent components obtained from curvature tensor

R00 = Ri0i0, (2.49)

R01 = R1001 +Ri0i1, (2.50)

R0i = R100i +Rj0ji, (2.51)

R11 = Ri1i1, (2.52)

R1i = R011i +Rj1ji, (2.53)

Rij = R1i0j +R0i1j +Rkikj, (2.54)

where we have used basic symmetries of curvature tensor

Rabcd = −Rbacd, (2.55)

Rabcd = −Rabdc, (2.56)

Rabcd = Rcdab = Rdcba. (2.57)


2.2.4 Other issues in NP formalism

Expressions given above are the most important equations of NP formalism in higher

dimensions, which we will use directly in our calculations in following chapters.

However, if we talk about generalization of NP formalism to higher dimensions we

should mention that this contains many other issues than we just mentioned here.

We found here just rotation coefficients, connection and curvatures form for null

congruence. To give complete generalization of NP formalism to higher dimensions

it means to find:

1. Transformation properties of rotation coefficients under transformations (2.21)-

(2.29), see [36].

2. Commutators of directional derivatives (2.36), see [35].

3. Ricci indetities, see [36].

4. Bianchi identities, see [34].

5. Classification of the Weyl tensor [37].

We refer reader for more details to given references and [38].

Chapter 3

Kerr-Schild spacetimes

Kerr-Schild (KS) spacetimes are very important in four dimensional general relativ-

ity. They include Kerr metric, which is one of the most important exact solutions

of Einstein’s equations in vacuum. Beside Kerr metric, KS spacetimes contain also

pp-waves and some Kundt spacetimes. Moreover electromagnetic field can be in-

cluded and so contain Kerr-Newman metric and pp-waves coupled to a null Maxwell

field. For references, see original paper [27] or the reference book [29]. Kerr-Schild

spacetimes also played a role in discovery of Mayer-Perry black holes in higher di-

mensions. Here we will try to find the generalization of the method [27] to higher

dimensions.

3.1 Kerr-Schild ansatz

We will study spacetime with metric given by Kerr-Schild ansatz

gab = ηab + 2Hkakb, (3.1)

where ηab is usual Minkowski metric in n dimensions, H is a scalar function and ka

is a null vector field

gabkakb = 0. (3.2)

From this we can see that −g = 1 and that ka is also null with respect to Minkowski

metric.

18

3.1. Kerr-Schild ansatz On higher dimensional Kerr-Schild spacetimes 19

Contravariant metric tensor is given by

gab = ηab − 2Hkakb. (3.3)

We choose null coordinates (u, v) in Minkowski spacetime which are related to Carte-

sian coordinates t, x1, ..xn−1 by

u =1√2(x1 − t),

v =1√2(x1 + t).

So then (3.1) can be expressed as

ds2 = 2dvdu+ dxidxi + 2Hk2. (3.4)

Now, we have to find the general field of real null directions in Minkowski spacetime.

We will find it to be 1 [35]

` = m1 = du− zidxi − 1

2z2dv, (3.5)

and the frame will be completed to be

n = m0 = dv +H`, (3.6)

mi = dxi + z idv. (3.7)

We will make the standard identification [29], [39]

` = k (3.8)

Frame directional derivatives (2.36) are acting on scalar function as

f|0 = Df = (∂v − zi∂i − 1

2z2∂u)f, (3.9)

f|1 = ∆f = (∂u −HD)f, (3.10)

f|i = δif = (∂i + zi∂u)f. (3.11)

It will turn out to be useful to have commutators of directional derivatives. In order

to find these commutators we first have to find commutators of D with all space-time

1Note, that zi and zi represents same quantity.

3.2. Rotation coefficients On higher dimensional Kerr-Schild spacetimes 20

directional derivatives.

∂vD −D∂v = −(∂vzi)δi, (3.12)

∂uD −D∂u = −(∂uzi)δi, (3.13)

∂jD −D∂j = −(∂jzi)δi. (3.14)

Now, we can find commutators of all frame directional derivatives, which are given

by

D∆−∆D = −H|0D + (∂uzi)δi = −H|0D + (zi|1 +Hzi|0)δi (3.15)

Dδi − δiD = zj |iδj + zi|0∂u = zj |iδj + zi|0(∆ +HD) (3.16)

∆δi − δi∆ = (∂uzi)∂u +H|iD −H(Dδi − δiD)

= zi|1(∆ +HD)−Hzj |iδj +H|iD (3.17)

δiδj − δjδi = −(zizj − zjzi)∂u∂u − (zj |i − zi|j)∂u

= 2z[j |i](∆ +HD). (3.18)

These commutators can be simplified, as we will see later by assuming null congru-

ence to be geodetic.

3.2 Rotation coefficients

In order to find the rotation coefficients we have to first find exterior derivatives of

frame vectors (2.14)

dn = H|ama ∧ `+Hd` = H|ama ∧ `+Hzi|ama ∧mi , (3.19)

d` = −zi|ama ∧ dxi − z izi|am

a ∧ dv = −zi|ama ∧mi , (3.20)

dmi = zi|ama ∧ dv = zi|am

a ∧ (n−H`). (3.21)

From this we can calculate the connection coefficients by using the first Cartan

equation (2.14)

d` = −Labma ∧mb , (3.22)

dn = −Nabma ∧mb , (3.23)

dmi = − i

M abma ∧mb . (3.24)


Here we should not forget that from the Cartan structure equation we can only

obtain antisymmetric part Γa

[bc]. Full expression for Γa

bcis given by equation (2.18).

In the following three subsections we will show detailed calculations of rotation

coefficients. For a reader who is not interested in these calculations we suggest to

skip to the last subsection 3.2.6, where all results will be listed.

3.2.1 Antisymmetric part of rotation coefficients

Let us first compute all antisymmetric parts of rotation coefficients from which we

will then compute all coefficients.

L[ab] : are found from equation (3.22) to be

zi|ama ∧mi = Labm

a ∧mb , (3.25)

→ L[01] = 0, (3.26)

L[0i] = zi|0, (3.27)

L[1i] = zi|1, (3.28)

L[ij] = zj |i − zi|j. (3.29)

N[ab] : are found from equation (3.23) to be

−H|ama ∧ `+Hzi|ama ∧mi = Nabm

a ∧mb, (3.30)

→ N[01] = −H|0, (3.31)

N[0i] = Hzi|0, (3.32)

N[1i] = Hzi|1 +H|i, (3.33)

N[ij] = Hzj |i −Hzi|j. (3.34)

i

M[ab] : are found from equation (3.24) to be

−zi|ama ∧ (n−H`) =

i

M abma ∧mb, (3.35)

→ i

M [01] = zi|1 +Hzi|0, (3.36)

i

M [0j] = zi|j, (3.37)

i

M [1j] = −Hzi|j, (3.38)

i

M [jk] = 0. (3.39)


3.2.2 Lab coefficients

Here we will determine all Lab using the knowledge of antisymmetric parts of rotation

coefficients and equation (2.18). Some of the rotation coefficients will be found o be

zero by constraints (2.34)

L00 = L01 = L0i = 0, (3.40)

L10 =1

2(L[10] − L[01]) = 0, (3.41)

L11 =1

2(N[10] −N[01]) = H|0, (3.42)

L1i =1

2(L[1i] +N[i0] −

i

M [01]) = 0, (3.43)

Li0 =1

2(L[i0] − L[0i]) = −zi|0, (3.44)

Li1 =1

2(L[i1] +

i

M [10] −N[0i]) = −zi|1 −Hzi|0 = ∂uzi, (3.45)

Lij =1

2(L[ij] +

i

M [j0] −j

M [0i]) = −zi|j. (3.46)

3.2.3 Nab coefficients

As in the previous section, we will obtain Nab from the knowledge of antisymmet-

ric parts of rotation coefficients and equation (2.18). Also some of the rotation

coefficients will be found to be zero by constraints (2.34) and we will simplify cal-

culations because we will be able to express some of Nab through Lab with the help

of constraints (2.35)

N10 = N11 = N1i = 0, (3.47)

N00 = −L10 = 0, (3.48)

N01 = −L11 = −H|0, (3.49)

N0i = −L1i = 0, (3.50)

Ni0 =1

2(N[i0] +

i

M [01] − L[1i]) = 0, (3.51)

Ni1 =1

2(N[i1] −N[1i]) = −H|i −Hzi|1, (3.52)

Nij =1

2(N[ij] +

i

M [j1] −j

M [1i]) = Hzj |i. (3.53)


3.2.4i

M ab coefficients

As in the previous two sections, we will obtaini

M ab from the knowledge of antisym-

metric parts of rotation coefficients and equation (2.18). Also some of the rotation

coefficents will be found to be zero by constraints (2.34) and we will simplify cal-

culations because we will be able to express some ofi

M ab through Lab andNab with

help of constraints (2.35). Indeed, these constraints will save us so much work, that

we will have to calculate just coeffcientsi

M ij andi

M ja

i

M i0 =i

M i1 =i

M ij = 0, (3.54)

i

M 00 = −Li0 = zi|0, (3.55)

i

M 01 = −Li1 = zi|1, (3.56)

i

M 0j = −Lij = zi|j, (3.57)

i

M 10 = −Ni0 = 0, (3.58)

i

M 11 = −Ni1 = +H|i +Hzi|1, (3.59)

i

M 1j = −Nij = −Hzj |i, (3.60)

i

M j0 =1

2(

i

M [j0] +j

M [0i] − L[ij]) = 0, (3.61)

i

M j1 =1

2(

i

M [j1] +j

M [1i] −N[ij]) = 2Hz[i|j], (3.62)

i

M jk = 0. (3.63)

3.2.5 Condition of geodecity

Physically the most interesting case is when the null congruence is geodetic. As it

was shown in the previous chapter in such a case matrix Lij has special geometric

meaning and characterize null congruence in an invariant way. Thus, we have to

prove that null congruence is geodetic in a case of Kerr-Schild spacetimes. In four

dimensions this is done for example in the standard reference book [29] and its

straightforward generalization to higher dimensions was given in [39].

Explicitly, Proposition 1 from [39] states that:

The null-vector ` in the Kerr-Schild metric (3.1) of an arbitrary dimen-


sion is geodetic, if and only if the energy-momentum tensor satisfies

Tabkakb = 0

This condition is clearly satisfied in a case of vacuum space-times, which will be of

our main interest here, but let us note that it is also satisfied in spacetimes with

cosmological constant and in the presence of matter fields aligned with the KS vector

`, for example Maxwell field.

Thus, proposition 1 from [39] is satisfied in space-times, which are of our interest

here and we can follow discussion in the previous chapter. We found that condition

of geodecity is equivalent to the vanishing of coefficient Li0 . In the previous cal-

culations we found thati

M 00 is given by this coefficient. Thus we can simplify our

calculations by setting both of these coefficients to zero

Li0 = − i

M 00 = 0. (3.64)

As a consequence of this we are obtaining constraint

zi|0 = 0, (3.65)

which will then simplify some other coefficients.

Note, that if ` is geodetic and

Ni0 =i

M j0 = 0 (3.66)

holds, then frame (3.5)-(3.7) is parallely transported along `.

Condition of geodecity will also simplify commutation relations of frame direc-

tional derivatives (3.15)-(3.18), which will be useful in the calculations of Riemann

tensor.

D∆−∆D = −H|0D + zi|1δi, (3.67)

Dδi − δiD = zj |iδj, (3.68)

∆δi − δi∆ = zi|1∆ + (H|i +Hzi|1)D −Hzj |iδj, (3.69)

δiδj − δjδi = 2z[j |i](∆ +HD). (3.70)

Note, that if we use our results for rotation coefficients, we can easily check that

these commutators are in agreement with the general results given in [35].


3.2.6 Summary

Here we can summarize all rotation coefficients found in previous calculations in a

clear way. We will use representation by tables, where rows are running first indix

and columns second one.

Lab b = 0 b = 1 b = j

a = 0 0 0 0

a = 1 0 H|0 0

a = i 0 −zi|1 −zi|j

Nab 0 1 j

0 0 −H|0 0

1 0 0 0

i 0 −H|i −Hzi|1 Hzj |i

i

M ab 0 1 k

0 0 zi|1 zi|k

1 0 Hzi|1 +H|i −Hzk|i

i 0 0 0

j 0 2Hz[i|j] 0

Table 3.1: Rotation coefficients Lab, Nab,i

M ab for geodetic congruence ` in the KS

spacetime (3.1) in the frame (3.5)-(3.7).

All these coefficients hold if condition of geodecity is satisfied, which is the most

interesting physical case, and with the help of which we were able to simplify our

calculations. In terms of functions zi this condition is

zi|0 = 0. (3.71)

3.3. Connection and curvature forms On higher dimensional Kerr-Schild spacetimes 26

3.3 Connection and curvature forms

From the knowledge of rotation coefficients we can now easily find the connection

forms which will be needed for finding curvature forms and consequently curvature

tensor. We have just these independent connection forms Γ01,Γ0i,Γ1i,Γij, which are

given by

Γ01 = −L1ama = N0am

a = −H|0`, (3.72)

Γ0i = −Liama =

i

M 0ama = zi|am

a = dzi, (3.73)

Γ1i = −Niama =

i

M 1ama = (H|i +Hzi|1)`−Hzj |im

j , (3.74)

Γij = − i

M jama =

j

M iama = −2Hz[i|j]`. (3.75)

Exterior derivatives of these connection forms are found to be

dΓ01 = −H|0|ama ∧ `+H|0zi|am

a ∧mi , (3.76)

dΓ0i = ddzi = 0, (3.77)

dΓ1i = [(H|i +Hzi|1)|a +H2zk|izk|a]ma ∧ `−Hzk|izk|am

a ∧ n− [(H|i +Hzi|1)zk|a + (Hzk|i)|a]m

a ∧mk , (3.78)

dΓij = −2(Hz[i|j])|ama ∧ `+ 2Hz[i|j]zk|am

a ∧mk , (3.79)

and corresponding curvature forms are given by equations (2.44)-(2.47).

1

2R01 = [H|0|a + zi|a(H|i +Hzi|1)]` ∧ma

+ (Hzk|azj|k +H|0zj|a)ma ∧mj , (3.80)

1

2R0i = (−H|0zi|a + 2Hzj|az[j |i])` ∧ma , (3.81)

1

2R1i = [(H|i +Hzi|1)|a +H2zk|izk|a]m

a ∧ `+Hzk|izk|an ∧ma

− [(H|i +Hzi|1)zk|a + (Hzk|i)|a]ma ∧mk

− (HH|0zk|i − 2H2z[i|j]zk|j)` ∧mk , (3.82)

1

2Rij = 2[(Hz[i|j])|a + (H[j +H∆z[j)zi]|a]` ∧ma

+ 2H(zk|[izj]|a − z[i|j]zk|a)mk ∧ma . (3.83)


3.3.1 Curvature Tensor

Now, we can easily find components of the Riemann tensor using relation

Rab = Rabcdmc ∧md . (3.84)

Independent components of Riemann tensor can be summarized in the following

table

R0101 −H|0|0

R010i 0

R011i H|0|i +H|0zi|1 +H|jzj |i +Hzj |izj|1 +Hzj|1zi|j

R01ij 2Hzk|[izj]|k + 2H|0z[j |i]

R0i0j 0

R0i1j −H|0zi|j − 2Hzk|jz[i|k]

R0ijk 0

R1i1j −(H|i +Hzi|1)|j −H2zk|izk|j − (H|i +Hzi|1)zj|1 − (Hzj |i)|1 −HH|0zj |i + 2H2z[i|k]zj|k

R1ijk 2(H|i +Hzi|1)z[j|k] + 2(Hδiz[k)|j]

Rijkl 2H(zk|[izj]|l] − zl|[izj]|k − 2z[i|j]z[k|l])

and other dependent components of curvature tensor can be summarized in the

following table

R0i01 0

R1i01 (H|i +Hzi|1)|0 +Hzk|izk|1

R1i0j Hzk|izk|j − (Hzj |i)|0

Rij01 −2(Hz[i|j])|0

Rij0k 0

Rij1k 2(Hz[i|j])|k + 2(H|[j +H∆z[j)zi]|k − 2H(zk|[izj]|l − z[i|j]zk|l)

Components in these two tables are related each to other by symmetry (2.57), i.e.

symmetry under exchange of pair of indices. For non-zero components these sym-

metries are

R1i01 = R011i, (3.85)

R1i0j = R0j 1i, (3.86)


Rij01 = R01ij, (3.87)

Rij1k = R1kij. (3.88)

To check this explicitly we need just commutators (3.67)-(3.70). It is a straight-

forward but tedious work, which would take a lot of place and thus we will not write

it in this thesis. To check the symmetries (2.55)-(2.56) (i.e. antisymmetry under

exchange of indices inside the first or second pair of indices) is trivial and thus only

symmetry which is left is as follows

R1i1j = R1j 1i. (3.89)

At the first sight this symmetry seems to not be satisfied, because R1i1j does not

seem to be symetric under exchange of indices i and j, but if we use commutators

(3.67)-(3.70) we will find that this symmetry holds. Thus Riemann tensor which we

have just found, satisfies all symmetries of Riemann tensor.

We can use it as a check of our calculations and also, as we can see, to simplify

our expressions for components for Riemann tensor, because some of the components

of Riemann tensor in the second table are in more compact form than in the first

one. Let us pick up each component in its most compact form from either of two

previous tables and all non-zero independent components can then be summarized

in the following table

R0101 −H|0|0

R011i (H|i +Hzi|1)|0 +Hzk|izk|1

R01ij 2(Hz[j |i])|0

R0i1j −H|0zi|j − 2Hzk|jz[i|k]

R1i1j −(δj + zj|1)(H|i +Hzi|1)−H2(zk|izk|j − 2z[i|k]zj|k)− (Hzj |i)|1 −HH|0zj |i

R1ijk 2(H|i +Hzi|1)z[j|k] + 2(Hδiz[k)|j]

Rijkl 2H(zk|[izj]|l − z[i|j]zk|l)

Table 3.2: Independent non-zero components of the curvature tensor

3.4. Einstein equations On higher dimensional Kerr-Schild spacetimes 29

3.4 Einstein equations

In order to write Einstein equations we have to find components of Ricci tensor

given by (2.49)-(2.54). Using the previous results can be found that they are given

by

R00 = R0i = 0, (3.90)

R01 = H|0|0 −H|0zi|i − 2Hzk|iz[i|k], (3.91)

R11 = −(δi + zi|1)(H|i +Hzi|1)−H2(zk|izk|i − 2z[i|k]zi|k) (3.92)

− (Hzi|i)|1 −HH|0zi|i, (3.93)

R1i = (H|i +Hzi|1)|0 +Hzk|izk|1 + 2(H|j +Hzj|1)z[i|j] + 2(Hδjz[i)|j], (3.94)

Rij = −2Hzi|kzj|k − 2(H|0 −Hzk|k)z(i|j). (3.95)

We can transform two of these components to more elegant and convenient form

using commutators (3.68)-(3.70)

R11 = −H|i|i − 2(Hzi|1)|i − 2Hzi|1zi|1 − (H|1 +HH|0)zi|i, (3.96)

R1i = (H|i +Hzi|1)|0 + 2(Hz[i|j])|j + 2H|[jzi]|j +Hzi|jzj|1. (3.97)

Now, we can summarize all independent non-zero components of Ricci tensor in

the following table If we use commutators (3.68)-(3.70) and expressions for rotation

R01 H|0|0 −H|0zi|i − 2Hzk|iz[i|k]

R11 −H|i|i − 2(Hzi|1)|i − 2Hzi|1zi|1 − (H|1 +HH|0)zi|i

R1i (H|i +Hzi|1)|0 + 2(Hz[i|j])|j + 2H|[jzi]|j +Hzi|jzj|1

Rij −2Hzi|kzj|k − 2(H|0 −Hzk|k)z(i|j)

Table 3.3: Non-zero components of Ricci tensor.

coefficients from the table 3.1, we can check whether our results are compatible

with [39]. After some straight-forward but tedious algebra we can see that our

results agree with the results of this paper if we make identification

H → −H,

3.4. Einstein equations On higher dimensional Kerr-Schild spacetimes 30

which is correct, because in [39] Kerr-Schild metric (3.1) with a minus sign is used,

while we are using a plus sign. Thus we can conclude that our results presented

here are fully consistent with [39]. With the knowledge of all components of Ricci

tensor we can write down vacuum Einstein’s equations for KS spacetimes (3.1) in

the frame (3.5)-(3.7) as follows

Rab = 0, (3.98)

where all Rab are given in the table 3.3. Solutions, or at least our attempts to solve

these equations, will be presented in the following chapter.

Chapter 4

Vacuum Kerr-Schild solutions

In preceding chapter we found vacuum Einstein’s equations for metric in a form of

the Kerr-Schilld ansatz (3.1). In this chapter we will try to solve these equations,

or at least to analyze them and show solutions in some special cases.

In a following calculations, values of optical scalars (expansion, shear and twist)

will play important role. Let us recall definitions of optical scalars (2.42) and express

them in terms of zi|j

Lij = −zi|j, (4.1)

θ = − 1

n− 2zi|i, (4.2)

σ2 = σijσij, (4.3)

ω2 = z[i|j]z[i|j], (4.4)

where shear matrix σij is

σij = −θδij − z(i|j). (4.5)

4.1 Vacuum Einstein’s equations

Einstein’s equations in vacuum can be found from results of previous chapter. They

are explicitly following equations, which we have rearranged according to increasing

31

4.2. Non-expanding solutions On higher dimensional Kerr-Schild spacetimes 32

complexity

Hzi|kzj|k + (H|0 −Hzk|k)z(i|j) = 0, (4.6)

H|0|0 −H|0zi|i − 2Hzk|iz[i|k] = 0, (4.7)

H|i|i + 2(Hzi|1)|i + 2Hzi|1zi|1 + (H|1 +HH|0)zi|i = 0, (4.8)

(H|i +Hzi|1)|0 + 2(Hz[i|j])|j + 2H|[jzi]|j +Hzi|jzj|1 = 0. (4.9)

Let us start our analysis with the first one. We can contract this equation with δij.

Note, that then this is equivalent to Einstein’s equation

Rii = 0,

to obtain

Hz2i|k +H|0zi|i −Hz2

i|i = 0, (4.10)

where we have used notation

z2i|k = zi|kzi|k, z2

i|i = (zi|i)2. (4.11)

From equation (4.10) we can find expression for H|0, but just under assumption that

the expansion, i.e. zi|i is non-zero. So, solutions with zero and nonzero expansion

have to be treated separately.

4.2 Non-expanding solutions

The case with zero expansion θ, i.e. zi|i = 0 is the simplest case, (4.10) implies that

zi|k = 0. (4.12)

Using this, we can find other Einstein’s equations to be

H|0|0 = 0, (4.13)

H|i|i + 2(Hzi|1)|i + 2Hzi|1zi|1 = 0, (4.14)

(H|i +Hzi|1)|0 = 0. (4.15)

4.3. Expanding solutions On higher dimensional Kerr-Schild spacetimes 33

With the help of commutators (3.68)-(3.70) we can find identities

zi|1|i = −zi|1zi|1, (4.16)

zi|1|0 = zi|0|1 = 0, (4.17)

H|i|0 = H|0|i, (4.18)

which simplify Einstein’s equations to the form

H|0|0 = 0, (4.19)

(δi + zi|1)H|0 = 0, (4.20)

(δi + 2zi|1)H|i = 0. (4.21)

Note, that this case was fully solved in [39], where it was found that non-expanding

KS spacetimes are equivalent to the Kundt solutions [40] of Weyl type N, which

belongs to the family of spacetimes with vanishing scalar invariants (VSI). Similiar

as in n = 4, in higher dimensional spacetimes Kundt solutions of Weyl type N consist

of two subfamilies: Kundt waves and pp-waves. For more detail discussion of these

solutions see [39], [40], [41].

4.3 Expanding solutions

In a case of non-zero expansion θ, i.e. zi|i 6= 0, we can express H|0 out of equation

(4.10) and obtain

H|0 = Hz2

i|i − z2i|k

zj|j. (4.22)

This can be inserted into the first Einstein’s equation (4.6), which will then take

form

zi|kzj|k =zl|kzl|kzm|m

z(i|j), (4.23)

which can be expressed in terms of rotation coefficients as

LikLjk =LlkLlk

(n− 2)θSij, (4.24)

what agrees with the result in [39].


It is important to observe that this condition on zi|j does not contain H and so it

is purely geometrical condition on the null congruence ` in the Minkowskian metric.

We will refer to it, in agreement with [39], as the optical constraint.

Because we have expression for H|0, we can try to simplify Einstein’s equation

(4.7) by substituting this expression there(Hz2

i|i − z2i|k

zj|j

)

|0−H|0zi|i − 2Hzk|iz[i|k] = 0, (4.25)

which we can simplify as

−H|0z2

i|kzj|j

+H(z2

i|i − z2i|k

zj|j

)

|0− 2Hzk|iz[i|k] = 0. (4.26)

The term in brackets is(z2

i|i − z2i|k

zj|j

)

|0=

(z2i|i + z2

i|k)zm|lzl|m − 2zi|kzm|kzi|mzj|jz2

j|j, (4.27)

and with the help of this and (4.22)we can simplify equation (4.25) to the following

formzi|kzi|kzj|lz(j|l)

z2m|m

=zi|kzj|kzi|jzm|m

. (4.28)

With the help of optical constraint we find out that this Einstein’s equation (4.7)

is in a case of expanding null congruence trivially satisfied .Unfortunately, it seems

that we are unable to simplify other Einstein’s equations significantly in the most

general case.

Before we continue our analyzes, let us summarize Einstein’s equations for ex-

panding null congruence `

zi|kzj|k =zl|kzl|kzm|m

z(i|j), (4.29)

H|0 = Hz2

i|i − z2i|k

zj|j, (4.30)

H|i|i + 2(Hzi|1)|i + 2Hzi|1zi|1 + (H|1 +HH|0)zi|i = 0, (4.31)

(H|i +Hzi|1)|0 + 2(Hz[i|j])|j + 2H|[jzi]|j +Hzi|jzj|1 = 0. (4.32)

We have to keep in mind that beside these equations, our null congruence also has

to satisfy the condition of geodecity

zi|0 = 0. (4.33)


4.3.1 Comments on solving Einstein’s equations

Our strategy in solving these equations can be divided into two steps:

1. At the beginning we should determine null congruence `. We can see that

the condition of geodecity (4.33) together with the optical constraint (4.29)

contains no terms with function H, i.e. these two equations determine the

form of null congruence `. Solution of these two equations should be the most

general null congruence admitted in a case of geodetic Kerr-Schild spacetime.

2. With the knowledge of most general admitted null congruence we can solve

rest of Einstein’s equations (4.30)-(4.32) in order to determine function H.

Let us just note that problem of finding most general null congruence admitted

by optical constraint and condition of geodecity, i.e. step 1., can be considered on

its own. It is analogue of Kerr theorem in a four dimensions, mentioned at the

end of section 1.2, and results are interesting even without solving other Einstein’s

equations.

Problem is that resulting equations (4.30)-(4.33) are extremely complicated and

idea to solve them analytically in general seems to be hopeless. If we will take a

look just on simplest of them, i.e. condition of geodecity (4.33), we can see big

difference from a four dimensional case. If we rewrite this equation in terms of

partial derivatives using (3.9)

(∂v − zj∂j − 1

2z2∂u)zi = 0, (4.34)

we can see that it is a system of quasilinear partial differential equations with non-

linear coupling. To find a general analytic solution is not possible, or at least

there exist no general procedure how to do it. In a four dimensions situations is

much simpler, condition of geodecity is just a quasilinear partial differential equation

Y|0 = 0 together with its complex conjugate [27], i.e. quasilinear PDE without non-

linear coupling. This is actually one of great advantages of using complex frame in

four dimensions, because then we have just one variable Y and solution is easy.

The optical constraint (4.29) is a system of non-linear partial differential equa-

tions, which also we do not know how to solve in fully general case. Rest of Einstein’s

equations (4.30)-(4.32) seems also hopeless to solve in full generality.

4.4. Non-twisting solutions On higher dimensional Kerr-Schild spacetimes 36

This means that we are forced to give up hope to find general Kerr-Schild solution

in a higher dimensions and we have to be satisfied with much less general and

complete solutions. There are three different ways how to proceed now in order to

find at least some Kerr-Schild solutions.

1. To try to analyze case when null congruence ` is non-twisting. We will see

that this will simplify our equations very much and we will be able to find

some restriction on solutions.

2. To try to guess some solutions which can be admitted by our equations. We

will find such ansatzes, which will be based either on some symmetry or on

Myers-Perry ansatz based on analogy with a four dimensional Kerr-Schild

solutions.

3. It seems that one of most general results which one can obtain is by method

started in [39]. This method is based on fixing some kind of r-dependency

along null congruence ` and investigating just r-dependency and asymptotic

behavior of Kerr-Schild solutions. This seems to be possible to do in fully

general case.

4.4 Non-twisting solutions

In this case we have ω = 0, i.e.

zi|j = zj |i. (4.35)

From what follows that

z(i|j) = zi|j z[i|j] = 0. (4.36)

Thus the optical constraint will take form

zi|kzj|k = Fzi|j. (4.37)

where

F =zl|kzl|kzm|m

. (4.38)

We can start our analysis of optical constraint with dividing our analysis into two

main categories according to degeneracy of the matrix zi|j.

4.5. Some explicit solutions On higher dimensional Kerr-Schild spacetimes 37

Non-degenerate zi|j

In a case of non-degenerate zi|j, i.e. det zi|j 6= 0, we can always find its inverse z−1

i|j .

Let us multiply with z−1

j|n the optical constraint (4.37) to obtain

zi|n = Fδin, (4.39)

i.e. that matrix zi|j is diagonal. Thus we proved that in a case of a non-degenerate

zi|j non-twisting solutions are also non-shearing. From [39] we know that only non-

shearing non-twisting solution is Schwarzschild-Tangherlini solution.

Degenerate zi|j

Solutions with det zi|j = 0 should represent black-string, according to [39]. For more

details and analyzes in general case consult [39]. In the following we will find some

ansatz, which will satisfy this condition and we will observe that it indeed represents

black string.

4.5 Some explicit solutions

As was mentioned in the section 4.3.1, to solve system (4.29)-(4.33) analytically in

a general case seems to be hopeless and as we could see that even in non-twisting

case we were not able to find general solution. What we can do now, is to try to

guess some solutions which would solve the system (4.29)-(4.33).

Solutions with dependency z(xi)

Our guess, when we are attempting to solve condition of geodecity (4.33) is to try

ansatz z(xi), i.e non-twisting solutions independent of coordinates u and v. Indeed,

in a such case condition of geodecity (4.33) will take simple form

zj∂jzi = 0. (4.40)

We may restrict ourselves to the case with zero twist, i.e.

∂izj = ∂jzi. (4.41)


If we combine these two conditions, by inserting second one to first one, we obtain

equation

∂izjzj ≡ ∂iz2 = 0, (4.42)

from where follows that

Null congruence given by ansatz zi =√

2xi

r

We can guess a particular solution

zi =√

2xi

r, (4.43)

where r is given by expression

r =√xixi =

√(x2)2 + (x3)2 + · · ·+ (xn−1)2. (4.44)

Note that

z2 = 2. (4.45)

We can derive important identities which we will use in following computation:

∂ir =zi√2, (4.46)

and

r∂r = xi∂i, (4.47)

from which it follows

zi∂i =√

2xi

r∂i =

√2∂r. (4.48)

We can easily find that directional derivatives of zi are

zi|0 = 0, (4.49)

zi|1 = 0, (4.50)

zi|j = ∂jzi =√

2δijr

2 − xixj

r3= zj |i. (4.51)

and from last one we can obtain following useful relations

zi|i =√

2n− 3

r, (4.52)

zi|jzi|j =2

r6(δijr

2 − xixj)(δijr2 − xixj) = 2

n− 3

r2. (4.53)

We can easily verify that optical constraint is satisfied. Determinant of zi|j is zero,

i.e. this solution corresponds to degenerate case mentioned in previous section.


Einstein’s equations for the above ansatz zi =√

2xi

r

Because (4.43) satisfies optical constraint, we can continue calculation of the rest of

Einstein’s equations in order to determine function H.

Einstein’s equation (4.30) will take form

H|0 =√

2H(n− 4)

r, (4.54)

Second Einstein’s equation (4.32) in a case of non-twisting null congruence ` is

H|i|0 + 2H|[jzi]|j = 0, (4.55)

which we can, with help of commutator (3.68), transform to the following form

H|0|i + 2H|jzi|j −H|izj|j = 0. (4.56)

Inserting previous result (4.54) we obtain

H|i −H(n− 4)√

2

zi

r−H|jzizj = 0 (4.57)

and remaining Einstein’s equation (4.31) become

H|i|i = −√

2∂uHn− 3

r(4.58)

Thus, equations (4.54)-(4.57) determine fully function H in a case of null congruence

given by (3.5) with (4.43). These equations contain directional derivatives along

frame vectors, so in order to solve (i.e. integrate) these equations we have to shift

to standard partial derivatives. Thus, H|i expressed through partial derivative will

be

H|i = ∂iH + zi∂uH. (4.59)

If we insert this to equation (4.57) we will obtain

∂iH− zi∂uH −H(n− 4)√2

zi

r−√

2zi∂rH = 0 (4.60)

We can multiply this equation by zi and we will get

√2∂uH = −∂rH−H(n− 4)

r, (4.61)


which we can insert back to equation (4.60) to obtain

∂iH =zi√2∂rH. (4.62)

Note, that, because of (4.46) , this equation tells us that dependence of function Hon xi is fully given by its dependence on r. Using previous two equations we can

find frame directional derivative (4.59)

H|i = −H(n− 4)√2

zi

r. (4.63)

Taking directional frame derivative of this expression we find

H|i|i = −(n− 4)√2

(∂i + zi∂u)[Hzi

r

]= −H(n− 4)2

r2− ∂rH(n− 4)

r−√

2∂uH(n− 4)

r,

(4.64)

and inserting previous result (4.61) we find that

H|i|i = 0, (4.65)

which allow us to simplify Einstein’s equation (4.58) to

∂uH = 0. (4.66)

Using (4.61) we can find frame derivative H|0 to be

H|0 = ∂vH−√

2∂rH− 1

2∂uH = ∂vH +

√2H(n− 4)

r, (4.67)

with this we can simplify Einstein’s equation (4.54) to

∂vH = 0. (4.68)

Thus Einstein’s equations (4.54)-(4.58) for congruence determined by (4.43) were

found to be

∂uH = 0, (4.69)

∂vH = 0, (4.70)

∂rH = −H(n− 4)

r. (4.71)

First two equations simply tell us that function H is independent of coordinates u

and v. According to (4.62) dependence ofH on xi is fully given by its dependency on


r. Thus H is function of only coordinate r and this dependence is fully determined

by third equation (4.71). We can easily integrate this equation and find that

H =M

r(n−4), (4.72)

where M is an arbitrary constant.

With use of (3.5) and (4.43) we can find that

` = (du− dv −√

2dr). (4.73)

Thus metric for ansatz (4.43) is given by

ds2BS = 2dvdu+ dxidx

i + 2M

r(n−4)

(du− dv +

√2dr

)2

. (4.74)

To understand this solution it is more instructive to make transformation of coor-

dinates. Using (3.4) null vector (4.73) in Cartesian coordinates will take form

` = −√

2(dt+ dr), (4.75)

then metric will be

ds2BS = −dt2 + (dx1)

2+ dr2 + r2dΩ2

n−3 + 4M

r(n−4)(dt+ dr)2 , (4.76)

where was used relation

dxidxi = dr2 + r2dΩ2

n−3. (4.77)

We can make trasformation

dt+ dr = dU, (4.78)

where U is some new coordinate and we will obtain

ds2BS = −

(1− Rg

r(n−4)

)dU2 + 2dUdr + r2dΩ2

n−3 + (dx1)2, (4.79)

where we redefined constant Rg = 4M . Using information from first chapter and [17]

we can easily see that first three terms of this metric corresponds to n−1 dimensional

Schwarzschild-Tangherlini solution in Eddington-Finkelstein coordinates. If we de-

note by ds2n−1(S) metric of n − 1 dimensional Schwarzschild-Tangherlini solution

then (4.79) we can rewrite as

ds2BS = ds2

n−1(S) + (dx1)2, (4.80)

from where we can see that it corresponds to black string [24], [13].

4.6. Fixing r-dependence On higher dimensional Kerr-Schild spacetimes 42

Generalization to black p-branes

In previous calculations we derived analytically black string solution by guessing

ansatz for functions zi in a form (4.43). We know that black string is topologically

S(n−1)×R, where S(n−1) represents (n−1) dimensional Schwarzchschild-Tangherlini

solution.

This solution is easy to generalize to black p-branes [13], which play important

role in string theory. Black p-branes are topologically S(n−p) ×Rp, i.e. black string

being just special name for black 1-brane.

Such generalization can be easily done by omitting (p − 1) coordinates from

ansatz (4.43) and definition of r-parameter (4.44). We can directly see that metric

in such case would be

ds2pB = −

(1− Rg

r(n−p−3)p

)dU2 + 2dUdrp + r2

pdΩ2n−p−2 +

p∑α=1

dxαdxα, (4.81)

where drp represents r-parameter defined as in (4.44), but with (p− 1) coordinates

omitted, i.e.

rp =n−1∑

α=p+1

√xαxα. (4.82)

We can rewrite this as

ds2pB = ds2

n−p(S) +

p∑α=1

dxαdxα, (4.83)

from where we can directly see that black p-brane is topologically S(n−p) ×Rp.

4.6 Fixing r-dependence

As we mentioned before, it is possible to choose some r dependence and analyze KS

spacetimes then. This is very useful mainly in analyzes of asymptotical properties of

the Riemann tensor. Natural such dependence is to choose r as a affine parameter

along null vector `, i.e. then

D =∂

∂r. (4.84)

This means that we have to make transform of coordinates

(u, v, xi) → (r, xA), (4.85)


where index A runs from 1 to n− 1.

In such coordinates other frame derivatives are given by

∆ = U∂

∂r+ Y A ∂

∂xA, (4.86)

and

δi = ωi

∂

∂r+ ξA

i

∂

∂xA. (4.87)

In order to analyze asymptotical properties of Riemann tensor (table 3.2), we have

to find asymptotical r-dependance of given directional derivatives, function H and

directional derivatives of functions zi.

We may start with analyzing r-dependance of directional derivatives of functions

zi. Condition of geodetecity (4.33) tells us that

zi|0 = ∂rzi = 0. (4.88)

To analyze r-dependance of zi|j we use commutator (3.68) and we obtain equation

∂rzi|j = zi|kzk|j. (4.89)

Note that this is equivalent to the Sachs equation used in [39].

Nondegenerate zi|j

Now, we focus on case when det zi|j 6= 0 and we can obtain

z−1

m|i∂rzi|j = zm|j. (4.90)

Here we can use identity

∂r(δij) = ∂r(z−1

i|k zk|j) = (∂rz−1

i|k )zk|j + z−1

i|k ∂rzk|j = 0, (4.91)

from where we can find that

z−1

i|k ∂rzk|j = −(∂rz−1

i|k )zk|j. (4.92)

If we insert this result back to (4.90) we obtain

(∂rz−1

m|i)zi|j = zm|j, (4.93)


from where it follows

∂rz−1

m|i = −δmi. (4.94)

We can easily integrate this to obtain

z−1

m|i = −rδmi + ami, (4.95)

where ami is matrix of functions with the property ∂rami = 0. We can easily invert

this equation under assumption that r goes to infinity

zi|j = −δijr

+aij

r2+O(r−3). (4.96)

Generalization of this result is the expansion [39], [42]

zi|j =k∑

p=1

(ap−1)ij

rp+O(r−p−1), (4.97)

where O(r−n) represents all terms which goes to zero as r−n and faster and (ap−1)ij

is p− 1 power of matrix aij.

With knowledge of (4.97), we can now easily find the r-dependance of zi|1. We

start with commutator (3.68), which if we apply on zi we obtain

∂rzi|1 = zi|jzj|1. (4.98)

We can multiply this equation by z−1

i|k and we obtain

z−1

i|k ∂rzi|1 = zi|1, (4.99)

and because of (4.95) we can use

∂rz−1

i|k = −δik, (4.100)

to simplify this equation to

∂r(z−1

i|k zi|1) = 0. (4.101)

Solution of this equation is

zi|1 = bjzi|j, (4.102)

where zi|j is defined by (4.97) and bi are some functions independent of r.


Using previous results we can find r-dependance of operators of directional

derivatives (4.86) and (4.87). In order to do so we have to use commutators (3.68)-

(3.70). Let us start with (4.87). Commutator (3.68) is given by

Dδi − δiD = zj |iδj, (4.103)

we can apply this on function r to obtain

∂rωi = zj |iωj. (4.104)

If we multiply this equation by z−1

i|k we obtain

z−1

i|k ∂rωi = ωk, (4.105)

and using (4.100) we can simplify it to

∂r(z−1

i|k ωi) = 0. (4.106)

This equation is solved by

ωi = ω0jzi|j, (4.107)

where ω0j

is some integration constant of equation (4.106).

Analogously, if we apply commutator (4.103) on coordinates xA we find that

ξAi

= ξAj

0zi|j, (4.108)

where ξAi

0are some integration constants. Thus we can write that r-dependance of

operator δi is given by

δi = zi|j(ω0j∂r + ξA

j

0∂A) (4.109)

In similar way we can find r-dependance of ∆ operator. Commutator (3.67) is given

by

D∆−∆D = −H|0D + zi|1δi. (4.110)

We can apply this commutator on r to obtain

∂rU = −∂rH + zi|1ωi, (4.111)

using result (4.107) we can rewrite this equation as

∂r(U +H) = ω0jzi|jzi|1, (4.112)


which we can with help of (4.98) simplify to

∂r(U +H) = ω0i∂rzi|1. (4.113)

Solution to this equation is easily found to be

U = U0 −H + ω0izi|1, (4.114)

where U0 is some function independent of r. Similarly, if we apply commutator

(4.110) on function xA we can obtain

Y A = Y A0+ ξA

i

0zi|1. (4.115)

Thus we can conclude that r dependance of operator ∆ is given by

∆ = (U0 −H + ω0izi|1)∂r + (Y A0

+ ξAi

0zi|1)∂A. (4.116)

Now, only remains to determine the r-dependance of KS function H. If we consider

just the leading term in expansion (4.97) then equation (4.30) has a simple form

∂rH = −H(n− 3)

r, (4.117)

which solution si given by

H =H0

rn−3, (4.118)

where H0 is function independent of r. Generalization of this result to arbitrary

term in r was already done in [39] and for non-degenerate case is given by

H =H0

rn−2q−3

q∏µ=1

1

r2 + (a0(2µ))

2, (4.119)

where H0 is function independent of r, 0 ≤ 2q ≤ n − 2. For more details about

meaning of functions (a0(2µ)) see [39].

Let us summarize previous results:

zi|j =k∑

p=1

(ap−1)ij

rp+O(r−p−1), (4.120)

zi|1 = bjzi|j, (4.121)

H =H0

rn−2q−3

q∏µ=1

1

r2 + (a0(2µ))

2, (4.122)


D = ∂r, (4.123)

∆ = U∂

∂r+ Y A ∂

∂xA, (4.124)

δi = ωi

∂

∂r+ ξA

i

∂

∂xA, (4.125)

where aij are matrices and H0 function independent of r, (a0(2µ)) are some constant

defined in [39], 0 ≤ 2q ≤ n− 2 and coefficients U , Y A, ωi and ξAi are given by

ωi = ω0jzi|j, (4.126)

ξAi

= ξAj

0zi|j, (4.127)

U = U0 −H + ω0izi|1, (4.128)

Y A = Y A0+ ξA

i

0zi|1, (4.129)

where ω0j, ξA

i

0, ω0

iand ξA

i

0are functions independent of r.

Now, we can start to analyze the properties of Riemann tensor given in table 3.3.

With the previous knowledge we can expand all obtained expressions just in terms

of zi|j and H. We will use shortcut ∂rH whenever it will be convenient, because we

know that

∂rH = Hz2

i|i − z2i|k

zj|j. (4.130)

Then Riemann tensor will be

R0101 = ∂r∂rH, (4.131)

R011i = (ω0j∂rH + ξA

j

0∂AH)zi|kzk|j + zi|j(ω

0j∂r∂rH + ξA

j

0∂r∂AH)

+ ∂rHzi|jbj + 2Hzi|jzj|kbk, (4.132)

R01ij = 2z[j |i]∂rH− 2H(zi|kzk|j − zj|kzk|i), (4.133)

R0i1j = −∂rHzi|j − 2Hzk|jz[i|k], (4.134)

R1i1j = −zj|k(ω0k∂r + ξA

k

0∂A)zi|l[(ω

0l∂r + ξA

l

0∂A)H] − zj|kzi|l(ω

0k∂rH + ξA

k

0∂AH)bl

− zi|kzj|l(ω0k∂rH + ξA

k

0∂AH)bl −Hzi|kzj|lblbk

− Hzj|k[(ω0k∂r + ξA

k

0∂A)zi|lbl]−H2(zk|izk|j − 2z[i|k]zj|k)−H(∂rH)zj |i

− (U0 −H− ω0kzk|lbl)zj |i∂rH− (U0 −H− ω0

kzk|lbl)Hzj|mzm|i

− ξAk

0zk|lbl∂A(zj |iH)− Y A0

∂A(zj |iH), (4.135)


R1ijk = 2(ω0l∂rH + ξA

l

0∂AH +Hbl)zi|lz[j|k]

+ (ω0l∂rH + ξA

l

0∂AH)(zj|lzk|i − zk|lzj |i) +H[zj|l(ω

0lzk|mzm|i + ξA

l

0∂Azk|i)

− zk|l(ω0lzj|mzm|i + ξA

l

0∂Azj |i)], (4.136)

Rijkl = 2H(zk|[izj]|l − z[i|j]zk|l). (4.137)

Here we found r-dependance of all components of Riemann tensor in general case.

This may be usefull in further calculations. For now, let us analyze only properties

of Riemann tensor at infinity. Because of previous results we know that to the

leading term holds

zi|j = O(r−1), (4.138)

H = O(rn−3). (4.139)

If we use these two relations we can find that all components of Riemann tensor

seems to be to the leading term in r

Rabcd = O(r1−n), (4.140)

except component R1i1j where last term seems to be of order O(r2−n). However, if

we take trace of this component we obtain

R11 = R1i1i. (4.141)

If we use Einstein’s equation R11 = 0 we can easily find that last

∂A(zi|iH) = O(r1−n). (4.142)

With this we proved that all components of Riemann tensor to the leading term are

Rabcd = O

(1

rn−1

). (4.143)

We can see that this result is in a agreement with general analyzes [42]. This

means that Kerr-Schild spacetimes do not ”peel off” and do not contain radiation

what is also in agreement with our knowldege about Kerr-Schild spacetimes in four

dimensions.

Conclusion

In this thesis we investigated Kerr-Schild spacetimes in higher (n > 4) dimensions.

For this we used a generalization of the Newman-Penrose formalism to higher di-

mensions. This has been developed only very recently and our work is thus one of

the first investigations that use it in its full strength.

We used special ansatz, so-called Kerr-Schild ansatz, where spacetime is fully

determined by null congruence ` and by a function H. For such ansatz we found

explicitly all Ricci rotation coefficients, connection forms and curvature forms. We

could observe that condition of ` to be geodetic is very important from both physical

viewpoint and viewpoint of calculations (it allow us to simplify many calculations).

Then using this knowledge we were able to find frame components of Riemann and

Ricci tensor. From there we found general form of gravitational part of Einstein’s

equations in higher dimensions for Kerr-Schild ansatz. These results were derived in

full generality and thus they may be useful in future studies of Kerr-Schild spacetimes

in higher dimensions. For example, in analyzes of solutions with electromagnetic

field or cosmological constant.

We restricted ourselves to the vacuum case and we proceeded to analyze Ein-

stein’s equations. We found that one of Einstein’s equations does not contain func-

tion H and together with condition of geodecity it determines fully null-congruence

`. Because of this we named this equation, in agreement with references, optical

constraint. Rest of Einstein’s equations determines unknown functions H.

We found that we have to distinguish two main classes of solutions, according

to value of expansion. Because non-expanding case is already well-studied and

understood, we focused on expanding solutions. In this case we found that one of

Einstein’s equations is trivially satisfied as a consequence of rest of equations.

We summarized remaining Einstein’s equations and we discussed possibility to

solve them. We found that situation in n > 4 dimensions is much more complicated

Conclusion On higher dimensional Kerr-Schild spacetimes 50

than 4 dimensional case, where general solutions exists. We could see that these

equations are systems of non-linear PDEs with nonlinear couplings. Because we

are unable to solve such systems we had to give up our hope for finding analytical

solution in general case.

Therefore, we argued that best what we can hope is to find some partial results.

We gave three possible ways how to proceed. First of them is to restrict ourselves

to the case of non-twisting `. We found that in such case we have to divide our

analyzes to two classes, according to degeneracy of matrix zi|j. We proved that, if

det(zi|j) 6= 0, then non-twisting solution is automatically non-shearing. We know

that the most general solution of this kind is Schwarzschild-Tangherlini. Solutions

with det(zi|j) = 0 were little discussed in general, according to earlier works this

case should include black strings.

Other way how to deal with impossibility to solve Einstein’s equations in general

case, was idea to guess some ansatz null-congruence `, which would satisfy these

equations and condition of geodecity. We restricted ourselves to the non-twisting

case and we gave some instructions how to guess such ansatz. We were successful and

we found one, which would satisfy condition of geodecity and optical constraint. We

solved rest of Einstein’s equations and determined function H in this special case.

We could see that this corresponds to black strings and with some modifications to

more general black p-branes. This is in agreement with previous analyzes, because

our ansatz for ` was satisfying condition det(zi|j) = 0 and thus black strings are

solution which we would expect.

Then we decided to fix r-dependence and analyze asymptotical properties of

Riemann tensor. We choosed affine parameter r along null vector `. We had to make

transformation of coordinates and we were able to find r-dependance of matrix zi|j

and zi|1 expressed as expansions. With knowledge of this and using properties of

commutators of operators of directional derivatives we could find r-dependance of

operators of directional derivatives.

With knowledge of this we could find expansions of all components of Riemann

tensor to arbitrary order in r. This result we obtained in generality and so it can be

usefull in future calculations. Because of complexity of Riemann tensor we decided

Conclusion On higher dimensional Kerr-Schild spacetimes 51

to ananalyze properties of Riemann tensor at infinity. We kept just leading terms in

all components and we were able to find that all components of Riemann tensor are

O(r1−n) to the leading term in r. This result is in an agreement with four dimensions

and its implication is that KS spacetimes does not contain gravitational radiation.

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Appendix A

Notation

Notation used here and in main references [27], [23], [39] are different from each

other. Here we adopted notation which is mostly combination of all of them and is

trying to be most instructive and clear. In following we give review of this notation

and also notation used in [27], [23], [39]. We hope that it will help the reader to

orientate faster here and in main references.

Metric, indices

Metric in a null frame herein is given by

gab =

0 1

1 0. . .

,

(where . . . represents diagonal matrix) while in [23] is difference in signs

gab =

0 −1

−1 0. . .

.

Space-time indices herein are written in Latin lower case, same as in [39], while

[27] and [23] are using Greek lower-case.

Frame or abstract indices are herein Latin lower case with hats. This nota-

tion is based on [43] and is very handful, because it makes clear difference between

56

Appendix A. Notation On higher dimensional Kerr-Schild spacetimes 57

frame and spacetime indices. In [39] frame indices are written usually in same way

as spacetime indices, just in case where authors want to stress difference are frame

indices written in round brackets. Other references [27] and [23] are using Latin

lower case for frame indices.

Beside this, reader should keep in mind that ranges of indices, according to place

of index in the alphabet, are as follows

a, b, · · · - Latin lower case from beginning of alphabet runs from 0 to n− 1.

i, j, · · · - Latin lower case from middle of alphabet runs from 2 to n− 1.

α, β, · · · - Greek lower case runs from 1 to bn−22c, where b· · ·c represents integer

part of given number.

Note, that these conventions about ranges of indices holds equally for frame and

space-time indices.

Symbols used

In following we give list of symbols used in this thesis and in main references. Note,

that some symbols used here do not have to have analogues in other references, this

denoted as ”−” in the table. Note, that we give list of corresponding symbols in main

references, but they do not necessarily describe same quantity, but corresponding

quantity. For example, functions zi used herein are not same functions as Y, Y used

in [27] in 4 dimensions, but they do play same role. Symbols are divided to categories

for easier orientation.

Metric:

Here [27] [23] [39] Description

n 4 N + 1 n number of dimensions

ηab ηµν ηµν ηab Minkowski metric

gab gµν gµν gab spacetime metric

gab gab gab gab frame metric


Frame, frame derivatives:

ma ea Ea m(a) covariant frame vectors

ma ea Ea m(a) contra-variant frame vectors

n e3 − n special symbol for m0

` e4 − ` special symbol for m1

ka kµ kµ ka Kerr-Schild null-vector, ` = k

zi, zi Y ,Y Ak − functions determining Kerr-Schild vector

H h H H Kerr-Schild function

δa − δa δa directional derivative along a-th frame vector

|a ,a − − short notation of directional derivative

D − D D directional derivative along 0-th frame vector

∆ − ∆ ∆ directional derivative along 1-th frame vector

Connection:

Γabc Γa

bc − − Ricci rotation coefficients

Nij − − Nij Γ0ij coefficient

Lij − − Lij Γ1ij coefficient

i

M jk − − i

M jk Γijk coefficient

Sij − − Sij symmetric part of Lij

Aij − − Aij anti-symmetric part of Lij, twist matrix

σij − σij shear matrix

θ θ − θ expansion scalar

σ 0 − σ shear scalar

ω ω − ω twist scalar

Γab Γab − − connection form

Curvature:

Rab Rab curvature form

Rabcd Rabcd Rabcd Rabcd frame components of curvature tensor

Cabcd − Cabcd Cabcd frame components of Weyl tensor

Rab Rab Rab Rab frame components of Ricci tensor


Other:

r r r r affine parameter, ”radius”

rp − − − affine parameter in n− p space dimensions. See sec. 4.5.

dΩn − − angular part of spherical metric in n dimenions

λ − − − boost weight, see sec. 2.2.

Xji − − − Spin matrix, SO(n− 2) see sec. 2.2.

Diploma Thesis - uni-bielefeld.dekrssak/pdf/Krssak... · 2013-03-19 · work on this thesis, mainly my parents and Olga. Declaration. I declare that I have written my diploma thesis

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